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October  2013, 33(10): 4567-4578. doi: 10.3934/dcds.2013.33.4567

An equivalent characterization of the summability condition for rational maps

Huaibin Li 1,

1. 

School of Mathematics and Information Science, Henan University, Kaifeng 475004, China

Received  October 2012 Revised  February 2013 Published  April 2013

We give an equivalent characterization of the summability condition in terms of the backward contracting property defined by Juan Rivera-Letelier, for rational maps of degree at least two which are expanding away from critical points.
Keywords: backward contracting property., Rational maps, summability condition.
Mathematics Subject Classification: Primary: 37F10; Secondary: 37F1.
Citation: Huaibin Li. An equivalent characterization of the summability condition for rational maps. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4567-4578. doi: 10.3934/dcds.2013.33.4567
References:
[1]

H. Bruin, J. Rivera-Letelier, W. Shen and S. van Strien, Large derivatives, backward contraction and invariant densities for interval maps, Invent. Math., 172 (2008), 509-533. doi: 10.1007/s00222-007-0108-4.  Google Scholar

[2]

H. Bruin, W. Shen and S. van Strien, Invariant measures exist without a growth condition, Comm. Math. Phys., 241 (2003), 287-306.  Google Scholar

[3]

P. Collet and J. Eckmann, Positive Liapunov exponents and absolute continuity for maps of the interval, Ergodic Theory Dynam. Systems, 3 (1983), 13-46. doi: 10.1017/S0143385700001802.  Google Scholar

[4]

J. Graczyk and S. Smirnov, Collet, Eckmann and Hölder, Invent. Math., 133 (1998), 69-96. doi: 10.1007/s002220050239.  Google Scholar

[5]

J. Graczyk and S. Smirnov, Non-uniform hyperbolicity in complex dynamics, Invent. Math., 175 (2009), 335-415. doi: 10.1007/s00222-008-0152-8.  Google Scholar

[6]

O. Kozlovski and S. van Strien, Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, Proc. Lond. Math. Soc. (3), 99 (2009), 275-296. doi: 10.1112/plms/pdn055.  Google Scholar

[7]

H. Li and W. Shen, Dimensions of rational maps satisfying the backward contraction property, Fund. Math., 198 (2008), 165-176. doi: 10.4064/fm198-2-6.  Google Scholar

[8]

H. Li and W. Shen, On non-uniform hyperbolicity assumptions in one-dimensional dynamics, Science China Math., 53 (2010), 1663-1677. doi: 10.1007/s11425-010-3134-4.  Google Scholar

[9]

H. Li and W. Shen, Topological invariance of a strong summability condition in one-dimensional dynamics, Int. Math. Res. Not., 8 (2013), 1783-1799. doi: 10.1093/imrn/rns105.  Google Scholar

[10]

T. Nowicki and D. Sands, Non-uniform hyperbolicity and universal bounds for S-unimodal maps, Invent. Math., 132 (1998), 633-680. doi: 10.1007/s002220050236.  Google Scholar

[11]

F. Przytycki, Hölder implies Collet-Eckmann, Géométrie complexe et systèes dynamiques (Orsay, 1995), Astérisque, 261 (2000), 385-403.  Google Scholar

[12]

F. Przytycki and J. Rivera-Letelier, Statistical properties of topological Collet-Eckman maps, Ann. Sci. Ecole Sup. Norm., 40 (2007), 135-178. doi: 10.1016/j.ansens.2006.11.002.  Google Scholar

[13]

F. Przytycki, J. Rivera-Letelier and S. Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Invent. Math., 151 (2003), 29-63. doi: 10.1007/s00222-002-0243-x.  Google Scholar

[14]

J. Rivera-Letelier, A connecting lemma for rational maps satisfying a no growth condition, Ergodic Theory Dynam. Systems, 27 (2007), 595-636. doi: 10.1017/S0143385706000629.  Google Scholar

[15]

J. Rivera-Letelier, Asymptotic expansion of smooth interval maps,, preprint, ().   Google Scholar

[16]

J. Rivera-Letelier and W. Shen, Statistical properties of one-dimensional maps under weak hyperbolicity assumptions,, preprint, ().   Google Scholar

show all references

References:
[1]

H. Bruin, J. Rivera-Letelier, W. Shen and S. van Strien, Large derivatives, backward contraction and invariant densities for interval maps, Invent. Math., 172 (2008), 509-533. doi: 10.1007/s00222-007-0108-4.  Google Scholar

[2]

H. Bruin, W. Shen and S. van Strien, Invariant measures exist without a growth condition, Comm. Math. Phys., 241 (2003), 287-306.  Google Scholar

[3]

P. Collet and J. Eckmann, Positive Liapunov exponents and absolute continuity for maps of the interval, Ergodic Theory Dynam. Systems, 3 (1983), 13-46. doi: 10.1017/S0143385700001802.  Google Scholar

[4]

J. Graczyk and S. Smirnov, Collet, Eckmann and Hölder, Invent. Math., 133 (1998), 69-96. doi: 10.1007/s002220050239.  Google Scholar

[5]

J. Graczyk and S. Smirnov, Non-uniform hyperbolicity in complex dynamics, Invent. Math., 175 (2009), 335-415. doi: 10.1007/s00222-008-0152-8.  Google Scholar

[6]

O. Kozlovski and S. van Strien, Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, Proc. Lond. Math. Soc. (3), 99 (2009), 275-296. doi: 10.1112/plms/pdn055.  Google Scholar

[7]

H. Li and W. Shen, Dimensions of rational maps satisfying the backward contraction property, Fund. Math., 198 (2008), 165-176. doi: 10.4064/fm198-2-6.  Google Scholar

[8]

H. Li and W. Shen, On non-uniform hyperbolicity assumptions in one-dimensional dynamics, Science China Math., 53 (2010), 1663-1677. doi: 10.1007/s11425-010-3134-4.  Google Scholar

[9]

H. Li and W. Shen, Topological invariance of a strong summability condition in one-dimensional dynamics, Int. Math. Res. Not., 8 (2013), 1783-1799. doi: 10.1093/imrn/rns105.  Google Scholar

[10]

T. Nowicki and D. Sands, Non-uniform hyperbolicity and universal bounds for S-unimodal maps, Invent. Math., 132 (1998), 633-680. doi: 10.1007/s002220050236.  Google Scholar

[11]

F. Przytycki, Hölder implies Collet-Eckmann, Géométrie complexe et systèes dynamiques (Orsay, 1995), Astérisque, 261 (2000), 385-403.  Google Scholar

[12]

F. Przytycki and J. Rivera-Letelier, Statistical properties of topological Collet-Eckman maps, Ann. Sci. Ecole Sup. Norm., 40 (2007), 135-178. doi: 10.1016/j.ansens.2006.11.002.  Google Scholar

[13]

F. Przytycki, J. Rivera-Letelier and S. Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Invent. Math., 151 (2003), 29-63. doi: 10.1007/s00222-002-0243-x.  Google Scholar

[14]

J. Rivera-Letelier, A connecting lemma for rational maps satisfying a no growth condition, Ergodic Theory Dynam. Systems, 27 (2007), 595-636. doi: 10.1017/S0143385706000629.  Google Scholar

[15]

J. Rivera-Letelier, Asymptotic expansion of smooth interval maps,, preprint, ().   Google Scholar

[16]

J. Rivera-Letelier and W. Shen, Statistical properties of one-dimensional maps under weak hyperbolicity assumptions,, preprint, ().   Google Scholar

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